Mollifiers are a concept that I missed in my prior math education; it is pleasant to encounter the concept now. Sometimes one has idea-knowledge about something, but there is not an attached vocabulary. The mental constructs, the objects and the ways they can be manipulated, do not have names. When someone else describes the objects and manipulations precisely and gives them names, suddenly one’s ideas become much clearer. Vocabulary facilitates thought. Thus it has been for me with the concept of a mollifier.
Briefly, a mollifier is a family of smoothing functions, consisting of localized blips. I will describe only in one dimension, real axis, but the concept extends to multiple dimensions. Suppose each blip function f_A to be non-zero only within distance A of the origin, non-negative, smooth, and have integral 1. You will recognize a family of approximations to the Dirac delta function. As A becomes smaller, approaching zero, f_A becomes more concentrated about the origin, but still maintains the area 1.
An example of a basis for a mollifier family is the blending function g() defined by g(x)=exp(-1/x) for x positive, zero otherwise. This is a smooth gradual rise, starting at x=0 and going to 1. Smoothness of g() at x=0 needs some proof, and I will give a reference below to a nice introductory article. Here is a graph of g(x) for x between 0 and 1.
The blending function g() starts very gradually at x=0; in fact smoothly, ie an infinitely number of derivatives exist at x=0 (and hence at all x values). As x goes to infinity, g(x) goes to 1, always monotonic. The important property of g() is that it blends smoothly from being always-zero for x negative, to becoming close to 1 as x gets large positive.
We wish to use the blending g() to define, given some positive parameter A, a smooth function f_A(z) which is suitable as one member of a mollifier family. That is, f_A(z) will be zero outside distance A from the origin, and have integral 1. The definition is f_A(z)=g(A-abs(z))/C_A where C_A is a constant chosen to make the integral of f_A equal 1. The A-abs(z) part will be negative for z outside the radius A from the origin. Taking g() of that provides a smooth transition on the boundary abs(z)=A. And dividing by C_A equal to the integral of g(A-abs(z)) normalizes f_A() to have integral 1.
You can see how to use the blending function g() to define other mollifier families, for instance one based upon g(A^2-abs(z)^2). In more than one dimension, for instance z=(z1,z2…zn), you can define abs(z) as the distance of the point z from the origin, hence abs(z)^2=z1^2+z2^2+…+zn^2. The important part, the core of the construct, is the availability of a smooth blending function g().
[OOPS — Hit Publish too soon. Will edit and close this off quickly. Return to topic in subsequent post.]
References: Article by John Loftin, Rutgers, about Mollifiers. Good introduction.
Wikipedia article about Mollifiers (pretty abstract).
http://en.wikipedia.org/wiki/Mollifier
Best wishes,
Ken Roberts
18-May-2014
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